\[{dN_1 \over dt} = r_1 N \left(1-{N_1 \over K_1} - \alpha {N_2 \over K_1}\right)\]

\[{dN_2 \over dt} = r_2 N \left(1-{N_2 \over K_2} - \beta {N_1 \over K_2}\right)\]

is the outcome of 2 density dependent phenomena:

1. Intra-specific competition (\(K_1\) and \(K_2\))
2. Inter-specific competition (\(\alpha\) and \(\beta\))

an ecological process where one organism (the predator) consumes another (the prey).

• Provides most of the principle route of energy flow through ecosystems
• Strong selective pressure
• Chief source of density dependent effects in regulation of many animal (and plant) populations

• Herbivory: animals eat plants
  • granivores: eat grain
  • frugivores: eat fruit
• Parasitism: animals eat other animals without (immediately) killing them
• Carnivory: animals eat other animals, killing them
• Cannibalism: animals eat their conspecifics

• Growth in predator population depends on the number of prey
• Growth in prey population depends on the number of predators
• As an extension of competition
• An extreme form of competition

Species 1 (competitor AND predator \(P\)) benefits:

\[{dP \over dt} = r_p P \left(1-{P \over K_p} - \alpha {V \over K_p}\right) + \gamma VP\]

Species 2 (competitor AND prey \(V\)) suffers:

\[{dV \over dt} = r_v V \left(1-{V \over K_v} - \beta { P\over K_v}\right) - \sigma V P\]

• \(\gamma\) (gamma) = conversion factor - how much does predators population benefit from eating prey?
• \(\sigma\) (sigma) = capture efficiency - how much does prey population suffer from getting eaten?

Equilibrium shifts UP toward the predator \(K_1\) and DOWN from prey \(K_2\).

That was complicated! Let’s simplify RADICALLY and eliminate ALL competition and density dependence (\(\alpha = \beta = 0\), \(K_v = K_p = \infty\)).

Lots! And mainly unrealistic!

• Standard continuous modeling stuff:
  • No age structure
  • Closed population
  • No time lag
• Instantaneous B & D responses of P to V and V to P
• Mass action: perfect population mixing in proportion to number of individuals

BUT it’s still a fun model to play with - and a good starting point.

\[{dV \over dt} = r V - \sigma VP\]

\(V\) - prey; \(P\) - predators; \(r\) - growth rate; \(\sigma\) - “capture efficiency”

No density dependence (\(K\)), No competition (\(\alpha N_2\)), only removal by \(P\) at rate \(\sigma\).

• Large \(\sigma\) - strong effect: each moose killed by each wolf has big impact
• Small \(\sigma\) - weak effect: bats and mosquitoes … hard to make an impact.

\[{dP \over dt} = - qP + \gamma VP \]

\(P\) - predators; \(V\) - prey; \(q\) - mortality rate; \(\gamma\) - “conversion” efficiency, e.g. how many prey = new predators

• High \(\gamma\): high dependency on each discrete prey item, e.g. anaconda and capybara
• Low \(\gamma\): low dependency on each discrete prey item, e.g. blue whale and krill

Based (mainly) on fur sales from the Hudson Bay Company in Canada over 100 years. Roughly a 9 to 11 year, fairly synchronous, cycle.

A theoretical wrinkle … add any carrying capacity to the model:

https://egurarie.shinyapps.io/predatorprey/ https://egurarie.shinyapps.io/isoclines/

… add any carrying capacity to the model:

And we lose all our oscillations!

Back to the very controlled laboratory environment …

Just put them together and see what happens

… Maybe one oscillation?

Dribble in 1 Paramecium and 1 Didinium into the dish every 3 days.

Conclusion: Oscillations are not intrinsic to predator prey dynamics, but require some local back-and-forth, immigration - emigration, local dynamics.